I want to make a few comments on Professor Feferman's "Does mathematics
need new axioms?". I haven't actually read the paper, but I've been to the
talk twice, and my comments are based on the talk.
The talk begins by noting that the question is vague. I agree--the
crucial vagueness lies in "need". Unfortunately, I recall no ensuing
attempt to sharpen the question.
Feferman's answer seems to be that mathematics doesn't even need the
axioms it has already, much less new ones. ( At least that's one
interpretation; the vagueness of the question has to be reflected in the
vagueness of the answer.) He argues that all scientifically applicable
mathematics can be developed in a system W which is conservative over PA,
and thus at the moment we need no more than W (for science? for
mathematics?). I don't know whether W is indeed sufficient for
scientifically applicable mathematics--the fact that some standard text in
functional analysis can be formalized within it is not enough--but even
granting this, does this mean we should stop developing the mathematics
outside W? It would require a massive re-education project to teach
humanity to live within W. Do we really want to teach our undergraduate
Analysis students to avoid using the full least upper bound property of
the reals? If some bright student uses it on his own, do we mark his paper
wrong? I asked these questions at one of the talks, and Feferman's answer
was that he did not advocate living within W, that it's just fine for
mathematics to go on using the nonconstructive existence principles it
currently does. ( I'm filling in a bit here. We never got to a discussion
of which of these principles are ok. 2nd order arithmetic? Zermelo?
ZFC? ZFC + an inaccesible?...?) But if we are not going to try to live
within W, what is its practical relevance to mathematical behavior?
Most people working in foam ( foundations of all mathematics, as
opposed to fopm, foundations of very important parts of mathematics)
aren't in it because they expect applications in Physics to come in their
lifetimes. One does seek applications in less theoretical areas, as close
to central and active topics as one can find them. In the case of large
cardinal axioms, one has the descriptive set theory of projective sets of
reals--and beyond--which they yield. In a more theoretical vein, one has
their role as a standard by which to calibrate consistency strengths.
There is a lot here, 30-35 years of continuous mathematical development
by a reasonably large number of people (by the standards of Logic). I
was disappointed that so little of this development made it into
Feferman's talks. A lay audience cannot judge the importance of this
theory rationally without being given a better idea of what it is.
Finally, Feferman points out that large cardinal hypotheses of the
sort we know cannot decide the CH (a result of Levy and Solovay), and
states his belief that the CH is not decided at all by the concept of set,
that it is in some sense "absolutely undecidable". I myself think the jury
is still out on this one. Again, the question is vague. If one wishes to
show that the CH is "absolutely undecidable", one needs a definition of
the term, just as one needed a definition of "effectively computable" in
order to show various functions are not effectively computable. If anyone
has even the beginnings of such a definition in the case of "absolutely
undecidable", I would be very interested to know the details. Of course,
those who say the CH can be decided by rational means bear a burden too,
namely, to do so. In this connection, it worth noting that we understand a
lot more about the CH than we did 30-35 years ago, and that large cardinal
hypotheses have played an important role in that increased understanding.
There are, for example, the forcing axioms (PFA and Woodin's P_max axiom)
which decide CH negatively ( even giving definable surjections of R onto
omega_2 in some cases), and Woodin's theorem that, roughly speaking, CH
acts as a "complete invariant" for the Sigma^2_1 theory of the reals,
granted the existence of a measurable Woodin cardinal. I think it is not
clear whether such results will lead toward a rational decision on the CH,
or toward the elucidation of a sense in which no such decision is
possible. Nevertheless, one ought to take them into account.
John Steel